Editing Characteristic (algebra)

{{Short description|Smallest integer n for which n equals 0 in a ring}}

In [[mathematics]], the '''characteristic''' of a [[ring (mathematics)|ring]] {{math|''R''}}, often denoted {{math|char(''R'')}}, is defined to be the smallest positive number of copies of the ring's [[identity element|multiplicative identity]] ({{math|1}}) that will sum to the [[additive identity]] ({{math|0}}). If no such number exists, the ring is said to have characteristic zero. 

That is, {{math|char(''R'')}} is the smallest positive number {{math|''n''}} such that:<ref name=Fraleigh-Brand-2020/>{{rp|style=ama|p= 198, Thm. 23.14}} 
: <math>\underbrace{1+\cdots+1}_{n \text{ summands}} = 0</math>
if such a number {{math|''n''}} exists, and {{math|0}} otherwise.

== Motivation ==
The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately.

The characteristic may also be taken to be the [[exponent (group theory)|exponent]] of the ring's [[additive group]], that is, the smallest positive integer {{math|''n''}} such that:<ref name=Fraleigh-Brand-2020>
{{cite book
 |first1=John B. |last1=Fraleigh
 |first2=Neal E. |last2=Brand
 |year=2020
 |title=A First Course in Abstract Algebra |edition=8th
 |publisher=[[Pearson Education]]
 |url=https://www.pearson.com/us/higher-education/program/Fraleigh-Pearson-e-Text-First-Course-in-Abstract-Algebra-A-Access-Card-8th-Edition/PGM282304.html
}}
</ref>{{rp|style=ama|p= 198, Def. 23.12}}
: <math>\underbrace{a+\cdots+a}_{n \text{ summands}} = 0</math>
for every element {{math|''a''}} of the ring (again, if {{math|''n''}} exists; otherwise zero). This definition applies in the more general class of [[Rng (algebra)|rng]]s (see ''{{slink|Ring (mathematics)#Multiplicative identity and the term "ring"}}''); for (unital) rings the two definitions are equivalent due to their [[distributive law]].

== Equivalent characterizations ==
* The characteristic of a ring {{math|''R''}} is the [[natural number]] {{math|''n''}} such that {{math|''n''<math>\mathbb{Z}</math>}} is the [[kernel (ring theory)|kernel]] of the unique [[ring homomorphism]] from <math>\mathbb{Z}</math> to {{math|''R''}}.{{efn|The requirements of ring homomorphisms are such that there can be only one (in fact, exactly one) homomorphism from the ring of integers to any ring; in the language of [[category theory]], <math>\mathbb{Z}</math> is an [[initial object]] of the [[category of rings]]. Again this applies when a ring has a multiplicative identity element (which is preserved by ring homomorphisms).}}
* The characteristic is the [[natural number]] {{math|''n''}} such that {{math|''R''}} contains a [[subring]] [[ring homomorphism|isomorphic]] to the [[factor ring]] <math>\mathbb{Z}/n\mathbb{Z}</math>, which is the [[image (mathematics)|image]] of the above homomorphism.
* When the non-negative integers {{math|{{mset|0, 1, 2, 3, ...}}}} are [[Partially ordered set|partially ordered]] by divisibility, then {{math|1}} is the smallest and {{math|0}} is the largest.  Then the characteristic of a ring is the smallest value of {{math|''n''}} for which {{math|''n'' &sdot; 1 {{=}} 0}}. If nothing "smaller" (in this ordering) than {{math|0}} will suffice, then the characteristic is&nbsp;{{math|0}}. This is the appropriate partial ordering because of such facts as that {{math|char(''A'' × ''B'')}} is the [[least common multiple]] of {{math|char ''A''}} and {{math|char ''B''}}, and that no ring homomorphism {{math|''f'' : ''A'' → ''B''}} exists unless {{math|char ''B''}} divides {{math|char ''A''}}.
* The characteristic of a ring {{math|''R''}} is {{math|''n''}} precisely if the statement {{math|''ka'' {{=}} 0}} for all {{math|''a'' ∈ ''R''}} implies that {{math|''k''}} is a multiple of {{math|''n''}}.

== Case of rings ==
If {{math|''R''}} and {{math|''S''}} are [[ring (mathematics)|rings]] and there exists a [[ring homomorphism]] {{math|''R'' → ''S''}}, then the characteristic of {{math|''S''}} divides the characteristic of {{math|''R''}}. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic {{math|1}} is the [[zero ring]], which has only a single element {{math|0}}. If a nontrivial ring {{math|''R''}} does not have any nontrivial [[zero divisor]]s, then its characteristic is either {{math|0}} or [[prime number|prime]]. In particular, this applies to all [[field (mathematics)|fields]], to all [[integral domain]]s, and to all [[division ring]]s. Any ring of characteristic zero is infinite.

The ring <math>\mathbb{Z}/n\mathbb{Z}</math> of integers [[modular arithmetic|modulo]] {{math|''n''}} has characteristic {{math|''n''}}. If {{math|''R''}} is a [[subring]] of {{math|''S''}}, then {{math|''R''}} and {{math|''S''}} have the same characteristic. For example, if {{math|''p''}} is prime and {{math|''q''(''X'')}} is an [[irreducible polynomial]] with coefficients in the field <math>\mathbb F_p</math> with {{mvar|p}} elements, then the [[quotient ring]] <math>\mathbb F_p[X]/(q(X))</math> is a field of characteristic {{math|''p''}}. Another example: The field <math>\mathbb{C}</math> of [[complex number]]s contains <math>\mathbb{Z}</math>, so the characteristic of <math>\mathbb{C}</math> is {{math|0}}.

A <math>\mathbb{Z}/n\mathbb{Z}</math>-algebra is equivalently a ring whose characteristic divides {{math|''n''}}. This is because for every ring {{math|''R''}} there is a ring homomorphism <math>\mathbb{Z}\to R</math>, and this map factors through <math>\mathbb{Z}/n\mathbb{Z}</math> if and only if the characteristic of {{math|''R''}} divides {{math|''n''}}. In this case for any {{math|''r''}} in the ring, then adding {{math|''r''}} to itself {{math|''n''}} times gives {{math|''nr'' {{=}} 0}}.

If a commutative ring {{math|''R''}} has ''prime characteristic'' {{math|''p''}}, then we have {{math|(''x'' + ''y''){{i sup|''p''}} {{=}} ''x''{{i sup|''p''}} + ''y''{{i sup|''p''}}}} for all elements {{math|''x''}} and {{math|''y''}} in {{math|''R''}} – the normally incorrect "[[freshman's dream]]" holds for power {{math|''p''}}.
The map {{math|''x'' ↦ ''x''{{i sup|''p''}}}} then defines a [[ring homomorphism]] {{math|''R'' → ''R''}}, which is called the ''[[Frobenius homomorphism]]''. If {{math|''R''}} is an [[integral domain]] it is [[injective]].

== Case of fields <span class="anchor" id="Fields"></span> ==
<!-- This section is an {{R to anchor}} from [[Characteristic exponent of a field]] and [[Prime characteristic]] -->
As mentioned above, the characteristic of any [[Field (mathematics)|field]] is either {{math|0}} or a prime number. A field of non-zero characteristic is called a field of ''finite characteristic'' or ''positive characteristic'' or ''prime characteristic''. The ''characteristic exponent'' is defined similarly, except that it is equal to {{math|1}} when the characteristic is {{math|0}}; otherwise it has the same value as the characteristic.<ref>
{{cite book
 | last = Bourbaki | first = Nicolas | author-link = Nicolas Bourbaki
 | contribution = 5. Characteristic exponent of a field. Perfect fields
 | contribution-url = https://books.google.com/books?id=GXT1CAAAQBAJ&pg=RA1-PA7
 | doi = 10.1007/978-3-642-61698-3
 | page = A.V.7
 | publisher = Springer
 | title = Algebra II, Chapters 4–7
 | year = 2003
| isbn = 978-3-540-00706-7 }}</ref>

Any field {{math|''F''}} has a unique minimal [[Field extension|subfield]], also called its [[prime field]]. This subfield is isomorphic to either the [[rational number]] field <math>\mathbb{Q}</math> or a finite field <math>\mathbb F_p</math> of prime order. Two prime fields of the same characteristic are isomorphic, and this isomorphism is unique. In other words, there is essentially a unique prime field in each characteristic.

=== Fields of characteristic zero ===
The fields of ''characteristic zero'' are those that have a subfield isomorphic to the field {{tmath|\Q}} of the [[rational number]]s. The most common of such fields are the subfields of the field {{tmath|\C}} of the [[complex number]]s; this includes the [[real number]]s <math>\mathbb{R}</math> and all [[algebraic number field]]s.

Other fields of characteristic zero are the [[p-adic field]]s that are widely used in number theory.

Fields of [[rational fraction]]s over the integers or a field of characteristic zero are other common examples.
 
[[Ordered field]]s always have characteristic zero; they include <math>\mathbb{Q}</math> and <math>\mathbb{R}.</math>

=== Fields of prime characteristic ===
The [[finite field]] {{math|GF(''p''{{sup|''n''}})}} has characteristic {{math|''p''}}.

There exist infinite fields of prime characteristic. For example, the field of all [[rational function]]s over <math>\mathbb{Z}/p\mathbb{Z}</math>, the [[algebraic closure]] of <math>\mathbb{Z}/p\mathbb{Z}</math> or the field of [[formal power series|formal Laurent series]] <math>\mathbb{Z}/p\mathbb{Z}((T))</math>. 

The size of any [[finite ring]] of prime characteristic {{math|''p''}} is a power of {{math|''p''}}. Since in that case it contains <math>\mathbb{Z}/p\mathbb{Z}</math> it is also a [[vector space]] over that field, and from [[linear algebra]] we know that the sizes of finite vector spaces over finite fields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power.{{efn|It is a vector space over a finite field, which we have shown to be of size {{math|''p''<sup>''n''</sup>}}, so its size is {{math|(''p''<sup>''n''</sup>)<sup>''m''</sup> {{=}} ''p''<sup>''nm''</sup>}}.}}

== See also ==
* [[Ring of mixed characteristic]]

== Notes ==
{{notelist}}

== References ==
{{reflist|25em}}

== Sources ==
{{wikibooks|Discrete Mathematics|Finite fields}}
{{refbegin}}
* {{cite book
   |first1=Neal H. |last1=McCoy
   |orig-year=1964 |year=1973
   |title=The Theory of Rings
   |publisher=[[Chelsea Publishing]]
   |page=4
   |isbn=978-0-8284-0266-8
  }}
{{refend}}

[[Category:Ring theory]]
[[Category:Field (mathematics)]]